As of July 2012, 777 exoplanets have been discovered utilizing mainly indirect detection techniques. The direct imaging of exoplanets is the next goal for astronomers, because it will reveal the diversity of planets and planetary systems, and will give access to the exoplanet's chemical composition via spectroscopy. With this spectroscopic knowledge, astronomers will be able to know, if a planet is terrestrial and, possibly, even find evidence of life. With so much potential, this branch of astronomy has also captivated the general public attention. The direct imaging of exoplanets remains a challenging task, due to (i) the extremely high contrast between the parent star and the orbiting exoplanet and (ii) their small angular separation. For ground-based observatories, this task is made even more difficult, due to the presence of atmospheric turbulence. High Contrast Imaging (HCI) instruments have been designed to meet this challenge. HCI instruments are usually composed of a coronagraph coupled with the full on-axis corrective capability of an Extreme Adaptive Optics (ExAO) system. An efficient coronagraph separates the faint planet's light from the much brighter starlight, but the dynamic boiling speckles, created by the stellar image, make exoplanet detection impossible without the help of a wavefront correction device. The Subaru Coronagraphic Extreme Adaptive Optics (SCExAO) system is a high performance HCI instrument developed at Subaru Telescope. The wavefront control system of SCExAO consists of three wavefront sensors (WFS) coupled with a 1024-actuator Micro-Electro-Mechanical-System (MEMS) deformable mirror (DM). MEMS DMs offer a large actuator density, allowing high count DMs to be deployed in small size beams. Therefore, MEMS DMs are an attractive technology for Adaptive Optics (AO) systems and are particularly well suited for HCI instruments employing ExAO technologies. SCExAO uses coherent light modulation in the focal plane introduced by the DM, for both wavefront sensing and correction. In this scheme, the DM is used to introduce known aberrations (speckles in the focal plane), which interfere with existing speckles. By monitoring the interference between the pre-existing speckles and the speckles added deliberately by the DM, it is possible to reconstruct the complex amplitude (amplitude and phase) of the focal plane speckles. Thus, the DM is used for wavefront sensing, in a scheme akin to phase diversity. For SCExAO and other HCI systems using phase diversity, the wavefront compensation is a mix of closed-loop and open-loop control of the DM. The successful implementation of MEMS DMs open-loop control relies on a thorough modelling of the DM response to the control system commands. The work presented in this thesis, motivated by the need to provide accurate DM control for the wavefront control system of SCExAO, was centred around the development of MEMS DM models. This dissertation reports the characterization of MEMS DMs and the development of two efficient modelling approaches. The open-loop performance of both approaches has been investigated. The model providing the best result has been implemented within the SCExAO wavefront control software. Within SCExAO, the model was used to command the DM to create focal plane speckles. The work is now focused on using the model within a full speckle nulling process and on increasing the execution speed to make the model suitable for on-sky operation.